[PPT] - On the relation between the magnitude and exponent of OTOCs (Based PowerPoint Presentation

SLIDE 1

On the relation between the magnitude and exponent of OTOCs

(Based on the paper with Alexei Kitaev [1812.00120])

Yingfei Gu

Harvard University

YITP, June 24, 2019

SLIDE 2

Outline

◮ Introduction: OTOC in SYK ◮ Kinetic equation approach: an algorithm that computes the

Lyapunov exponent

◮ Ladder identity and branching time ◮ Applications of the ladder identity

SLIDE 3

Introduction

◮ OTOCs in large N systems with all-to-all interaction: t OTOC β tscr

Early time: A(t)B(0)A(t)B(0)connected ∼ 1 N eλLt C

◮ Convention β = 2π. MSS chaos bound 0 < λL 1. ◮ Main example: SYK HSYK =

jklm Jjklmχjχkχlχm

SLIDE 4

OTOC in SYK

◮ At strong coupling J ≫ 1, SYK is almost maximally chaotic (Maldacena-Stanford, 2016)

OTOC(t) ∼ 1 N eλLt C , λL ∼ 1 − c1 J , 1 C ∼ c2J

◮ Prefactor 1 C is big: pseudo-Goldstone mode (Schwarzian

mode).

◮ The ratio 1−λL C

has a finite limit at J → ∞.

◮ This talk is about an identity relating λL to C. [Corollary: although c1, c2 individually depends on the UV data, the product c1c2 is universal.] ◮ Hope: improve the understanding of fast scrambling.

SLIDE 5

Four point function and kinetic equation

Average over fermion indices, in general depends on four times. OTOC(t1, t2

future

, t3, t4

past

) = 1 N2

jk

χj(t1)χk(t3)χj(t2)χk(t4) + G(t12)G(t34)

◮ At large N limit, four-point functions are dominated by ladder

diagrams (Kitaev, Polchinski-Rosenhaus, Maldacena-Stanford ...) + + + . . .

◮ Euclidean four-point function: Bethe-Salpeter equation

F =

n

Fn, Fn = · Fn−1 ⇒ F = F0 + KF

◮ OTOC: deform the contour to double Keldysh ⇒ Kinetic equation

OTOC ≈ K R · OTOC, K R(t1, t2, t3, t4) :

R R W 1 2 3 4

SLIDE 6

Single-mode ansatz

How does OTOC(t1, t2

future

, t3, t4

past

) look like?

◮ We are interested in time regime t1 ≈ t2 ≫ t3 ≈ t4: large

separation between the perturbations in the past and probes in the future. (But still before the scrambling time.)

◮ Single-mode ansatz (“scramblon”) (Kitaev-Suh, 2017)

OTOC(t1, t2, t3, t4) ≈ 1 N eλL(t1+t2−t3−t4)/2 C Y R(t12)Y A(t34)

R A 1 2 3 4

Y R/A : vertex function

SLIDE 7

Eigenvalue kR(α)

The kinetic equation OTOC ≈ K R OTOC means OTOC is an eigenvector of K R with eigenvalue kR = 1. In general finding OTOC is a complicated search. We can reduce it to a shooting problem:

◮ Guess a growing exponent α for scramblon eαt; ◮ Adjust vertex function such that the trial OTOC(α) is an

eigenvector of K R with general eigenvalue kR(α);

◮ Find α∗ such that kR(α∗) = 1 ⇒ λL = α∗.

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Prefactor

This algorithm is useful in finding λL (as well as vertex functions), but the prefactor is ambiguous

◮ Because the kinetic equation OTOC ≈ K R OTOC is a

homogeneous linear equation

◮ Origin of the ambiguity: we dropped the constant term F0 in

the exact equation OTOC = K R OTOC +F0

◮ We would like to get the correct prefactor with the data

(λL, Y R/A, kR(α)) we already have

SLIDE 9

Ladder identity

OTOC(t1, t2, t3, t4) ≈ 1 N eλL(t1+t2−t3−t4)/2 C Y R(t12)Y A(t34)

◮ Ladder identity:

2 cos λLπ

2

C · (−k′

R(λL)) ·

Y A, Y R

= 1 .

◮

Y A, Y R : inner product of vertex functions:

Y A, Y R

= = (q − 1)J2

dt Y A(t)
G W (t)

q−2Y R(t) .

◮ Branching time tB := −k′

R(λL). Average separation between

adjacent rungs. tB

t2 t1 t4 t3

SLIDE 10

Idea of the derivation

Idea: cut a long ladder into pieces and find a consistency condition.

◮ Cut in the middle, find adjacent t5+t6

2

< t0 < t7+t8

2

t1 t2 t3 t4 t0 t5 t6 t7 t8

◮ Cut-gluing consistency condition 1 2 3 4

≈

1 2

· ·

3 4 ◮ Left hand side contains one copy of C −1, right hand side has two

copies of C −1.

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Applications

The ladder identity: 2 cos λLπ

2

C · tB ·

Y A, Y R

= 1 . Applications:

◮ Computational shortcuts: C ⇔ λL; ◮ In a 1D model, exact maximal chaos.

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Computational shortcuts

◮ C ⇒ λL: for SYK at strong coupling, the prefactor C is determined

by the coefficient of the Schwarzian action αS. However, the Lyapunov exponent will receive small correction from the conformal matters (Maldacena-Stanford, 2016) δλL ≈ 6αS Jk′

R(−1)∆(1 − ∆)(1 − 2∆) tan(π∆) .

◮ λL ⇒ C: for q-body interacting SYK model at q → ∞ limit, the

exact value of λL is easy to find using the kinetic equation

(Maldacena-Stanford, 2016),

λL 2 cos πλL

2

= J . But the prefactor is hard to compute. The ladder identity gives a result consistent with the computation using Liouville equation

(Qi-Streicher, 2018).

OTOC(t1, t2; t3, t4) ≈ 1 N cos λLπ

2

eλL(t1+t2−t3−t4)/2

2 cosh λLt12

2

2 cosh λLt34

2

.

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Maximal chaos in a 1D model

◮ Idea: regard λL and C as analytic functions of some parameter,

2 cos λLπ

2

C · tB ·

Y A, Y R

= 1 . then the analytical properties of λL and C are locked by the ladder identity.

◮ A concrete example: SYK chain (YG-Qi-Stanford, 2016)

k j

J′

jklm,x m l k l j m

Jjklm,x−1

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Operators at two different locations

Spatial propagation of chaos: measured by A(x, t)B(0, 0)A(x, t)B(0, 0)

x t t = x/vB

◮ Fourier transform:

OTOC(x, t) = dp 2π eipx OTOC(p, t)

◮ Each OTOC(p, t) can be solved using retarded kernel approach.

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Fourier Transform

◮ The ladder identity holds for each OTOC(p, t):

C(p) = 2 cos λL(p)π 2 · tB · (Y A, Y R) ,

◮ The dependence of tB and (Y A, Y R) on p is not important

(analytic and do not vanish in the domain of interest). OTOC(x, t) ∼ 1 N +∞

−∞

dp 2π eλL(p)t+ipx 2 cos πλL(p)

2

u(x,t)

· Y RY A tB(Y A, Y R) .

SLIDE 16

Butterfly wavefront

u(x, t) = +∞

−∞

dp 2π eλL(p)t+ipx 2 cos λL(p)

2

◮ Butterfly wavefront: u(x, t) ∼ 1. ◮ For large x > 0 and t, we can estimate the asymptotics by saddle

point of the exponent. λ′

L(p)t + ix = 0,

p = i|p|.

◮ The relevant saddle point ps = i|ps| is purely imaginary: deform the

integral contour to pass. Warning: pole contribution! cos λL(p1)π 2 = 0, λL(p1) = 1, p1 = i|p1| .

SLIDE 17

Pole contribution: maximal chaos

◮ When J > Jc, the pole dominates the butterfly wavefront:

u1(x, t) ≈ et−|p1||x| πiλ′

L(p1),

v1 = 1 |p1|

◮ Four regions with different OTOC behavior.

x t t = x/vB t = x/vB + tscr t = x/v∗ 1 2 3 4

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Summary and discussion

An identity relates C and λL; 2 cos λLπ

2

C · tB ·

Y A, Y R

= 1 , tB = (−k′

R(λL))

Some lessons from the applications:

◮ Computational shortcuts: in SYK model we find the correction

δλL ∝ t−1

B . If we call λL = 1 the “coherent scrambling”

(Schwarzian, gravity...), then the branching time tB measures the “decoherence” effect.

◮ Maximal chaos: spatial locality provides a mechanism to enhance

the Lyapunov exponent. Ladder identity: the enhancement cancels the correction exactly

Thank you!

SLIDE 19

Derivation: cos λLπ

2

factor

◮ Naively, we would have a formula:

OTOC ≈ OTOCL · BOX · OTOCR

◮ Subtlety: multiple choices on the double Keldysh contour

t1 t2 t3 t4 t0 t5 t6 t7 t8

◮ Sum of two choices:

OTOC ≈

ei

λLπ 2

+ e−i

λLπ 2

OTOCL · BOX · OTOCR